Integrally small perturbations of semigroups and stability of partial differential equations (Q457877)

Summary: Let \(A\) be a generator of an exponentially stable operator semigroup in a Banach space, and let \(C(t)\) \((t\geq 0)\) be a linear bounded variable operator. Assuming that \(\int^t_0C(s)ds\) is sufficiently small in a certain sense for the equation \(dx/dt=Ax+C(t)x\), we derive exponential stability conditions. Besides, we do not require that for each \(t_0\geq 0\), the ``frozen'' autonomous equation \(dx/dt=Ax+C(t_0)x\) is stable. In particular, we consider evolution equations with periodic operator coefficients. These results are applied to partial differential equations.